Republican Donald Trump and Democrat Bernie Sanders scored decisive victories in Tuesday’s New Hampshire primary – as Ohio Gov. John Kasich surged into a second-place finish and Florida Sen. Marco Rubio was fighting to stay in the race with Ted Cruz and Jeb Bush for third.

After months of campaigning in the snow, the battle for the South now begins -- and the runners-up will be looking for new life, and in some cases survival, as the winners celebrate. 

“We are going to make America great again,” the victorious Trump told cheering supporters Tuesday night. “We are going now to South Carolina. We are gonna win in South Carolina!” 

Chris Christie, though, is staying behind -- returning to New Jersey after a disappointing sixth-place finish, planning to make a decision by Wednesday on the way forward with the viability of his campaign now in doubt.  

The rest head to the Palmetto State, each facing a unique challenge: Trump aims to hold his front-runner status after cementing it in New Hampshire on the heels of his Iowa loss. Kasich, while getting a big boost out of Tuesday’s contest, will look to demonstrate he can play beyond New Hampshire. And as Cruz and Bush look to build on their performance, Rubio is vowing to improve after a rocky debate seemingly disrupted his momentum.

"That will never happen again," Rubio told supporters Tuesday, referring to his Saturday debate performance.

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On the Democratic side, meanwhile, Hillary Clinton is facing the prospect of losing her front-runner status if Sanders can close his polling gap in South Carolina and beyond.

The Vermont senator voiced confidence at his victory rally Tuesday night.

“Tonight we serve notice to the political and economic establishment of this country,” Sanders said. “Now it’s on to Nevada, South Carolina and beyond.” 

The Clinton campaign tried to cast the candidate’s primary loss in the best light, saying in a memo that the outcome was “long expected” and prospects for future victories remain “very good.”

“I want to say I still love New Hampshire and I always will,” Clinton told supporters at a Tuesday night rally. “Now we take this campaign to the entire country. We are going to fight for every vote in every state.” 

For the winners of both contests Tuesday night, New Hampshire was a comeback after each came in second last week in Iowa. 

Trump won decisively; incoming returns show him with 35 percent, towering over the field. And Sanders was crushing Clinton 60-39 percent. 

Kasich clocked in second with about 16 percent. 

The battle for third, though, remained a fierce contest among three candidates – Texas Sen. Cruz, former Florida Gov. Bush and Rubio. Cruz had 12 percent, Bush had 11 percent and Rubio had 10 percent, according to the latest returns.

Fox News can project that former HP CEO Carly Fiorina will finish seventh, and retired neurosurgeon Ben Carson will finish eighth. 

For Kasich, his time in the state appeared to pay off and the second-place finish gives a major boost to his campaign, especially after he finished a distant eighth in Iowa last week.

It also dealt a blow to Rubio, whose strong, third-place finish in Iowa eight days ago gave him momentum that lasted until Saturday, when Rubio stumbled in the debate amid withering attacks from Christie. 

The closeness of the race for third, fourth and fifth places still might end up prolonging the Republican nominating contest because it complicates the establishment’s efforts to coalesce around a single consensus candidate.

Yet the person who tripped up Rubio in Saturday’s debate, Christie, was mired in single digits in Tuesday's race, leaving him to weigh his options back home. 

Exit polls suggest Trump was aided in large part in the state by independents. They showed 36 percent of the state’s influential independents were backing Trump, leading the field among that bloc. 

On the Republican side, Trump led virtually every poll in the state in the run-up to the contest. 

And on the Democratic side, Clinton consistently trailed Sanders going into Tuesday’s contest. But the margin of victory for both Sanders and Trump was substantial. 

Clinton herself has pushed back on reports that the campaign may be looking at a shake-up in the near future – but the speculation is sure to mount following her Granite State loss.

From here, the campaigns head next to Nevada and South Carolina, which hold the next two contests before Super Tuesday at the beginning of March.